p-group, metabelian, nilpotent (class 3), monomial
Aliases: M5(2).19C22, (C2×D4).6C8, (C2×Q8).6C8, C8.124(C2×D4), (C2×C8).390D4, C8○(C23.C8), C23.C8⋊8C2, C23.4(C2×C8), (C2×M5(2))⋊9C2, (C22×C8).20C4, C8.63(C22⋊C4), C4.15(C22⋊C8), (C2×C8).385C23, (C2×C4).24M4(2), C4.50(C2×M4(2)), (C2×M4(2)).30C4, C22.5(C22⋊C8), C22.13(C22×C8), (C22×C8).416C22, (C2×M4(2)).329C22, (C2×C4).5(C2×C8), (C2×C8).250(C2×C4), (C2×C4○D4).18C4, (C2×C8○D4).17C2, C2.25(C2×C22⋊C8), C4.116(C2×C22⋊C4), (C22×C4).77(C2×C4), (C2×C4).559(C22×C4), (C2×C4).267(C22⋊C4), SmallGroup(128,847)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M5(2).19C22
G = < a,b,c,d | a16=b2=c2=d2=1, bab=a9, cac=ab, ad=da, bc=cb, bd=db, dcd=a8c >
Subgroups: 172 in 110 conjugacy classes, 58 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C2×C16, M5(2), M5(2), C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, C23.C8, C2×M5(2), C2×C8○D4, M5(2).19C22
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), C2×C22⋊C8, M5(2).19C22
►On 32 points
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)
(2 10)(3 11)(6 14)(7 15)(17 25)(20 28)(21 29)(24 32)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)
G:=sub<Sym(32)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32), (2,10)(3,11)(6,14)(7,15)(17,25)(20,28)(21,29)(24,32), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32), (2,10)(3,11)(6,14)(7,15)(17,25)(20,28)(21,29)(24,32), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32)], [(2,10),(3,11),(6,14),(7,15),(17,25),(20,28),(21,29),(24,32)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | D4 | M4(2) | M5(2).19C22 |
| kernel | M5(2).19C22 | C23.C8 | C2×M5(2) | C2×C8○D4 | C22×C8 | C2×M4(2) | C2×C4○D4 | C2×D4 | C2×Q8 | C2×C8 | C2×C4 | C1 |
| # reps | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 12 | 4 | 4 | 4 | 4 |
Matrix representation of M5(2).19C22 ►in GL4(𝔽17) generated by
| 6 | 0 | 15 | 0 |
| 10 | 0 | 0 | 1 |
| 1 | 16 | 11 | 0 |
| 10 | 0 | 3 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 6 | 0 | 16 | 0 |
| 14 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 3 | 0 | 0 | 1 |
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 5 | 14 | 0 | 9 |
| 11 | 10 | 2 | 0 |
G:=sub<GL(4,GF(17))| [6,10,1,10,0,0,16,0,15,0,11,3,0,1,0,0],[1,0,6,14,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,3,0,1,0,0,0,0,16,0,0,0,0,1],[0,16,5,11,16,0,14,10,0,0,0,2,0,0,9,0] >;
M5(2).19C22 in GAP, Magma, Sage, TeX
M_5(2)._{19}C_2^2 % in TeX
G:=Group("M5(2).19C2^2"); // GroupNames label
G:=SmallGroup(128,847);
// by ID
G=gap.SmallGroup(128,847);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,2019,1411,102,124]);
// Polycyclic
G:=Group<a,b,c,d|a^16=b^2=c^2=d^2=1,b*a*b=a^9,c*a*c=a*b,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^8*c>;
// generators/relations